Triangle calculator SSA

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Triangle has two solutions with side c=65.35223498236 and with side c=3.25992554143

#1 Obtuse scalene triangle.

Sides: a = 37   b = 34   c = 65.35223498236

Area: T = 452.9066291074
Perimeter: p = 136.3522349824
Semiperimeter: s = 68.17661749118

Angle ∠ A = α = 24.05879314829° = 24°3'29″ = 0.42198901156 rad
Angle ∠ B = β = 22° = 0.38439724354 rad
Angle ∠ C = γ = 133.9422068517° = 133°56'31″ = 2.33877301026 rad

Height: ha = 24.48114211392
Height: hb = 26.64215465338
Height: hc = 13.86604439564

Median: ma = 48.69551210465
Median: mb = 50.309869521
Median: mc = 13.95659160621

Inradius: r = 6.64331754445
Circumradius: R = 45.38109417634

Vertex coordinates: A[65.35223498236; 0] B[0; 0] C[34.3065802619; 13.86604439564]
Centroid: CG[33.21993841475; 4.62201479855]
Coordinates of the circumscribed circle: U[32.67661749118; -31.49112284369]
Coordinates of the inscribed circle: I[34.17661749118; 6.64331754445]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 155.9422068517° = 155°56'31″ = 0.42198901156 rad
∠ B' = β' = 158° = 0.38439724354 rad
∠ C' = γ' = 46.05879314829° = 46°3'29″ = 2.33877301026 rad




How did we calculate this triangle?

Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 37 ; ; b = 34 ; ; c = 65.35 ; ;

1. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 37+34+65.35 = 136.35 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 136.35 }{ 2 } = 68.18 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 68.18 * (68.18-37)(68.18-34)(68.18-65.35) } ; ; T = sqrt{ 205124.11 } = 452.91 ; ;

4. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 452.91 }{ 37 } = 24.48 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 452.91 }{ 34 } = 26.64 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 452.91 }{ 65.35 } = 13.86 ; ;

5. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 37**2-34**2-65.35**2 }{ 2 * 34 * 65.35 } ) = 24° 3'29" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 34**2-37**2-65.35**2 }{ 2 * 37 * 65.35 } ) = 22° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 65.35**2-37**2-34**2 }{ 2 * 34 * 37 } ) = 133° 56'31" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 452.91 }{ 68.18 } = 6.64 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 37 }{ 2 * sin 24° 3'29" } = 45.38 ; ;





#2 Obtuse scalene triangle.

Sides: a = 37   b = 34   c = 3.25992554143

Area: T = 22.58773635047
Perimeter: p = 74.25992554143
Semiperimeter: s = 37.13296277072

Angle ∠ A = α = 155.9422068517° = 155°56'31″ = 2.7221702538 rad
Angle ∠ B = β = 22° = 0.38439724354 rad
Angle ∠ C = γ = 2.05879314829° = 2°3'29″ = 0.03659176802 rad

Height: ha = 1.22109385678
Height: hb = 1.32986684415
Height: hc = 13.86604439564

Median: ma = 15.52661512593
Median: mb = 20.02202740473
Median: mc = 35.49442856462

Inradius: r = 0.60883380012
Circumradius: R = 45.38109417634

Vertex coordinates: A[3.25992554143; 0] B[0; 0] C[34.3065802619; 13.86604439564]
Centroid: CG[12.52216860111; 4.62201479855]
Coordinates of the circumscribed circle: U[1.63296277072; 45.35216723933]
Coordinates of the inscribed circle: I[3.13296277072; 0.60883380012]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 24.05879314829° = 24°3'29″ = 2.7221702538 rad
∠ B' = β' = 158° = 0.38439724354 rad
∠ C' = γ' = 177.9422068517° = 177°56'31″ = 0.03659176802 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 37 ; ; b = 34 ; ; beta = 22° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 34**2 = 37**2 + c**2 -2 * 34 * c * cos (22° ) ; ; ; ; c**2 -68.612c +213 =0 ; ; p=1; q=-68.6116052379; r=213 ; ; D = q**2 - 4pr = 68.612**2 - 4 * 1 * 213 = 3855.55237333 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 68.61 ± sqrt{ 3855.55 } }{ 2 } ; ; c_{1,2} = 34.305802619 ± 31.0465472047 ; ; c_{1} = 65.3523498236 ; ;
c_{2} = 3.2592554143 ; ; ; ; (c -65.3523498236) (c -3.2592554143) = 0 ; ; ; ; c>0 ; ;


Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 37 ; ; b = 34 ; ; c = 3.26 ; ;

2. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 37+34+3.26 = 74.26 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 74.26 }{ 2 } = 37.13 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 37.13 * (37.13-37)(37.13-34)(37.13-3.26) } ; ; T = sqrt{ 510.19 } = 22.59 ; ;

5. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 22.59 }{ 37 } = 1.22 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 22.59 }{ 34 } = 1.33 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 22.59 }{ 3.26 } = 13.86 ; ;

6. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 37**2-34**2-3.26**2 }{ 2 * 34 * 3.26 } ) = 155° 56'31" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 34**2-37**2-3.26**2 }{ 2 * 37 * 3.26 } ) = 22° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 3.26**2-37**2-34**2 }{ 2 * 34 * 37 } ) = 2° 3'29" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 22.59 }{ 37.13 } = 0.61 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 37 }{ 2 * sin 155° 56'31" } = 45.38 ; ;




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